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只关注循环执行次数最多的一段代码
总复杂度等于最高阶项的复杂度
嵌套代码的复杂度等于嵌套内外代码复杂度的乘积

二、排序算法1、冒泡排序1.1、基本介绍冒泡排序（Bubble Sorting）的基">
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                <h1 id="数据结构预算法"><a href="#数据结构预算法" class="headerlink" title="数据结构预算法"></a>数据结构预算法</h1><h2 id="一、算法复杂度"><a href="#一、算法复杂度" class="headerlink" title="一、算法复杂度"></a>一、算法复杂度</h2><h3 id="1、时间复杂度"><a href="#1、时间复杂度" class="headerlink" title="1、时间复杂度"></a>1、时间复杂度</h3><ul>
<li>只关注循环执行次数最多的一段代码</li>
<li>总复杂度等于最高阶项的复杂度</li>
<li>嵌套代码的复杂度等于嵌套内外代码复杂度的乘积</li>
</ul>
<h2 id="二、排序算法"><a href="#二、排序算法" class="headerlink" title="二、排序算法"></a>二、排序算法</h2><h3 id="1、冒泡排序"><a href="#1、冒泡排序" class="headerlink" title="1、冒泡排序"></a>1、冒泡排序</h3><h4 id="1-1、基本介绍"><a href="#1-1、基本介绍" class="headerlink" title="1.1、基本介绍"></a>1.1、基本介绍</h4><p>冒泡排序（Bubble Sorting）的基本思想是：通过对待排序序列从前向后（从下标较小的元素开始）,依次比较相邻元素的值，若发现逆序则交换，使值较大的元素逐渐从前移向后部，就象水底下的气泡一样逐渐向上冒<br><strong>优化：</strong><br>因为排序的过程中，各元素不断接近自己的位置，如果一趟比较下来没有进行过交换，就说明序列有序，因此要在排序过程中设置一个标志 flag 判断元素是否进行过交换。从而减少不必要的比较。(这里说的优化，可以在冒泡排<br>序写好后，在进行)</p>
<h4 id="1-2、演示冒泡过程"><a href="#1-2、演示冒泡过程" class="headerlink" title="1.2、演示冒泡过程"></a>1.2、演示冒泡过程</h4><p><img src="https://notes-hub.oss-cn-beijing.aliyuncs.com/typora_images/image-20220904210640366.png" alt="image-20220904210640366"></p>
<h4 id="1-3、冒泡排序应用实例"><a href="#1-3、冒泡排序应用实例" class="headerlink" title="1.3、冒泡排序应用实例"></a>1.3、冒泡排序应用实例</h4><blockquote>
<p>我们举一个具体的案例来说明冒泡法。我们将五个无序的数：3, 9, -1, 10, -2 使用冒泡排序法将其排成一个从小到大的有序数列。</p>
</blockquote>
<p><strong>coding：</strong></p>
<pre class=" language-java"><code class="language-java"><span class="token keyword">public</span> <span class="token keyword">class</span> <span class="token class-name">BubbleSorting</span> <span class="token punctuation">{</span>
    <span class="token keyword">static</span> <span class="token keyword">int</span> arr<span class="token punctuation">[</span><span class="token punctuation">]</span> <span class="token operator">=</span> <span class="token punctuation">{</span><span class="token number">3</span><span class="token punctuation">,</span> <span class="token number">9</span><span class="token punctuation">,</span> <span class="token operator">-</span><span class="token number">1</span><span class="token punctuation">,</span> <span class="token number">10</span><span class="token punctuation">,</span> <span class="token operator">-</span><span class="token number">2</span><span class="token punctuation">}</span><span class="token punctuation">;</span>

    <span class="token keyword">public</span> <span class="token keyword">static</span> <span class="token keyword">void</span> <span class="token function">main</span><span class="token punctuation">(</span>String<span class="token punctuation">[</span><span class="token punctuation">]</span> args<span class="token punctuation">)</span> <span class="token punctuation">{</span>
        <span class="token function">sort</span><span class="token punctuation">(</span>arr<span class="token punctuation">)</span><span class="token punctuation">;</span>
    <span class="token punctuation">}</span>
    <span class="token keyword">public</span> <span class="token keyword">static</span> <span class="token keyword">void</span> <span class="token function">sort</span><span class="token punctuation">(</span><span class="token keyword">int</span><span class="token punctuation">[</span><span class="token punctuation">]</span> arr<span class="token punctuation">)</span> <span class="token punctuation">{</span>
        <span class="token keyword">for</span> <span class="token punctuation">(</span><span class="token keyword">int</span> j <span class="token operator">=</span> <span class="token number">0</span><span class="token punctuation">;</span> j <span class="token operator">&lt;</span> arr<span class="token punctuation">.</span>length<span class="token punctuation">;</span> j<span class="token operator">++</span><span class="token punctuation">)</span> <span class="token punctuation">{</span>
            <span class="token keyword">for</span> <span class="token punctuation">(</span><span class="token keyword">int</span> i <span class="token operator">=</span> <span class="token number">1</span><span class="token punctuation">;</span> i <span class="token operator">&lt;</span> arr<span class="token punctuation">.</span>length<span class="token punctuation">;</span> i<span class="token operator">++</span><span class="token punctuation">)</span> <span class="token punctuation">{</span>
                <span class="token keyword">if</span> <span class="token punctuation">(</span>arr<span class="token punctuation">[</span>i <span class="token operator">-</span> <span class="token number">1</span><span class="token punctuation">]</span> <span class="token operator">></span> arr<span class="token punctuation">[</span>i<span class="token punctuation">]</span><span class="token punctuation">)</span> <span class="token punctuation">{</span>
                    <span class="token keyword">int</span> temp <span class="token operator">=</span> arr<span class="token punctuation">[</span>i <span class="token operator">-</span> <span class="token number">1</span><span class="token punctuation">]</span><span class="token punctuation">;</span>
                    arr<span class="token punctuation">[</span>i <span class="token operator">-</span> <span class="token number">1</span><span class="token punctuation">]</span> <span class="token operator">=</span> arr<span class="token punctuation">[</span>i<span class="token punctuation">]</span><span class="token punctuation">;</span>
                    arr<span class="token punctuation">[</span>i<span class="token punctuation">]</span> <span class="token operator">=</span> temp<span class="token punctuation">;</span>
                <span class="token punctuation">}</span>
            <span class="token punctuation">}</span>
        <span class="token punctuation">}</span>
        <span class="token keyword">for</span> <span class="token punctuation">(</span><span class="token keyword">int</span> i <span class="token operator">=</span> <span class="token number">0</span><span class="token punctuation">;</span> i <span class="token operator">&lt;</span> arr<span class="token punctuation">.</span>length<span class="token punctuation">;</span> i<span class="token operator">++</span><span class="token punctuation">)</span> <span class="token punctuation">{</span>
            System<span class="token punctuation">.</span>out<span class="token punctuation">.</span><span class="token function">println</span><span class="token punctuation">(</span>arr<span class="token punctuation">[</span>i<span class="token punctuation">]</span><span class="token punctuation">)</span><span class="token punctuation">;</span>
        <span class="token punctuation">}</span>
    <span class="token punctuation">}</span>
<span class="token punctuation">}</span>
</code></pre>
<h3 id="2、选择排序"><a href="#2、选择排序" class="headerlink" title="2、选择排序"></a>2、选择排序</h3><p>选择排序（select sorting）也是一种简单的排序方法。它的基本思想是：第一次从 arr[0]<del>arr[n-1]中选取最小值，与 arr[0]交换，第二次从 arr[1]</del>arr[n-1]中选取最小值，与 arr[1]交换，第三次从 arr[2]<del>arr[n-1]中选取最小值，与 arr[2]交换，…，第 i 次从 arr[i-1]</del>arr[n-1]中选取最小值，与 arr[i-1]交换，…, 第 n-1 次从 arr[n-2]~arr[n-1]中选取最小值，与 arr[n-2]交换，总共通过 n-1 次，得到一个按排序码从小到大排列的有序序列。</p>
<h4 id="2-1、选择排序思路分析图"><a href="#2-1、选择排序思路分析图" class="headerlink" title="2.1、选择排序思路分析图"></a>2.1、选择排序思路分析图</h4><p><img src="https://notes-hub.oss-cn-beijing.aliyuncs.com/typora_images/image-20220904212626441.png" alt="image-20220904212626441"></p>
<h4 id="2-2、选择排序应用实例"><a href="#2-2、选择排序应用实例" class="headerlink" title="2.2、选择排序应用实例"></a>2.2、选择排序应用实例</h4><blockquote>
<p>有一群牛 , 颜值分别是 101, 34, 119, 1 请使用选择排序从低到高进行排序 [101, 34, 119, 1]</p>
</blockquote>
<p><strong>coding：</strong></p>
<pre class=" language-java"><code class="language-java"><span class="token keyword">public</span> <span class="token keyword">class</span> <span class="token class-name">SelectSorting</span> <span class="token punctuation">{</span>
    <span class="token keyword">static</span> <span class="token keyword">int</span> arr<span class="token punctuation">[</span><span class="token punctuation">]</span> <span class="token operator">=</span> <span class="token punctuation">{</span><span class="token number">101</span><span class="token punctuation">,</span><span class="token number">34</span><span class="token punctuation">,</span><span class="token number">119</span><span class="token punctuation">,</span><span class="token number">1</span><span class="token punctuation">}</span><span class="token punctuation">;</span>

    <span class="token keyword">public</span> <span class="token keyword">static</span> <span class="token keyword">void</span> <span class="token function">main</span><span class="token punctuation">(</span>String<span class="token punctuation">[</span><span class="token punctuation">]</span> args<span class="token punctuation">)</span> <span class="token punctuation">{</span>
        <span class="token function">sort</span><span class="token punctuation">(</span>arr<span class="token punctuation">)</span><span class="token punctuation">;</span>
    <span class="token punctuation">}</span>
    <span class="token keyword">public</span> <span class="token keyword">static</span> <span class="token keyword">void</span> <span class="token function">sort</span><span class="token punctuation">(</span><span class="token keyword">int</span><span class="token punctuation">[</span><span class="token punctuation">]</span> arr<span class="token punctuation">)</span><span class="token punctuation">{</span>


        <span class="token keyword">for</span> <span class="token punctuation">(</span><span class="token keyword">int</span> i <span class="token operator">=</span> <span class="token number">0</span><span class="token punctuation">;</span> i <span class="token operator">&lt;</span> arr<span class="token punctuation">.</span>length<span class="token punctuation">;</span> i<span class="token operator">++</span><span class="token punctuation">)</span> <span class="token punctuation">{</span>
            <span class="token keyword">int</span> minIndex <span class="token operator">=</span> i<span class="token punctuation">;</span>
            <span class="token keyword">int</span> min <span class="token operator">=</span> arr<span class="token punctuation">[</span>i<span class="token punctuation">]</span><span class="token punctuation">;</span>
            <span class="token keyword">for</span> <span class="token punctuation">(</span><span class="token keyword">int</span> j <span class="token operator">=</span> i<span class="token operator">+</span><span class="token number">1</span><span class="token punctuation">;</span> j <span class="token operator">&lt;</span> arr<span class="token punctuation">.</span>length<span class="token punctuation">;</span> j<span class="token operator">++</span><span class="token punctuation">)</span> <span class="token punctuation">{</span>
                <span class="token keyword">if</span> <span class="token punctuation">(</span>arr<span class="token punctuation">[</span>j<span class="token punctuation">]</span><span class="token operator">&lt;</span>min<span class="token punctuation">)</span><span class="token punctuation">{</span>
                    min <span class="token operator">=</span> arr<span class="token punctuation">[</span>j<span class="token punctuation">]</span><span class="token punctuation">;</span>
                    minIndex <span class="token operator">=</span> j<span class="token punctuation">;</span>
                <span class="token punctuation">}</span>
            <span class="token punctuation">}</span>
            <span class="token keyword">if</span> <span class="token punctuation">(</span>minIndex<span class="token operator">!=</span>i<span class="token punctuation">)</span><span class="token punctuation">{</span>
                arr<span class="token punctuation">[</span>minIndex<span class="token punctuation">]</span> <span class="token operator">=</span> arr<span class="token punctuation">[</span>i<span class="token punctuation">]</span><span class="token punctuation">;</span>
                arr<span class="token punctuation">[</span>i<span class="token punctuation">]</span> <span class="token operator">=</span> min<span class="token punctuation">;</span>
            <span class="token punctuation">}</span>
        <span class="token punctuation">}</span>
        <span class="token keyword">for</span> <span class="token punctuation">(</span><span class="token keyword">int</span> i <span class="token operator">=</span> <span class="token number">0</span><span class="token punctuation">;</span> i <span class="token operator">&lt;</span> arr<span class="token punctuation">.</span>length<span class="token punctuation">;</span> i<span class="token operator">++</span><span class="token punctuation">)</span> <span class="token punctuation">{</span>
            System<span class="token punctuation">.</span>out<span class="token punctuation">.</span><span class="token function">println</span><span class="token punctuation">(</span>arr<span class="token punctuation">[</span>i<span class="token punctuation">]</span><span class="token punctuation">)</span><span class="token punctuation">;</span>
        <span class="token punctuation">}</span>
    <span class="token punctuation">}</span>
<span class="token punctuation">}</span>
</code></pre>
<h3 id="3、插入排序"><a href="#3、插入排序" class="headerlink" title="3、插入排序"></a>3、插入排序</h3><p>插入排序（Insertion Sorting）的基本思想是：把 n 个待排序的元素看成为一个有序表和一个无序表，开始时有序表中只包含一个元素，无序表中包含有 n-1 个元素，排序过程中每次从无序表中取出第一个元素，把它的排序码依次与有序表元素的排序码进行比较，将它插入到有序表中的适当位置，使之成为新的有序表</p>
<h4 id="3-1、插入排序思路图"><a href="#3-1、插入排序思路图" class="headerlink" title="3.1、插入排序思路图"></a>3.1、插入排序思路图</h4><p><img src="https://notes-hub.oss-cn-beijing.aliyuncs.com/typora_images/image-20220904215211378.png" alt="image-20220904215211378"></p>
<h4 id="3-2、插入排序法应用实例"><a href="#3-2、插入排序法应用实例" class="headerlink" title="3.2、插入排序法应用实例"></a>3.2、插入排序法应用实例</h4><blockquote>
<p>有一群小牛, 考试成绩分别是 101, 34, 119, 1 请从小到大排序</p>
</blockquote>
<p><strong>coding：</strong></p>
<pre class=" language-java"><code class="language-java"><span class="token keyword">public</span> <span class="token keyword">class</span> <span class="token class-name">InsertSorting</span> <span class="token punctuation">{</span>
    <span class="token keyword">static</span> <span class="token keyword">int</span><span class="token punctuation">[</span><span class="token punctuation">]</span> arr <span class="token operator">=</span> <span class="token punctuation">{</span><span class="token number">101</span><span class="token punctuation">,</span> <span class="token number">34</span><span class="token punctuation">,</span> <span class="token number">119</span><span class="token punctuation">,</span> <span class="token number">1</span><span class="token punctuation">}</span><span class="token punctuation">;</span>

    <span class="token keyword">public</span> <span class="token keyword">static</span> <span class="token keyword">void</span> <span class="token function">main</span><span class="token punctuation">(</span>String<span class="token punctuation">[</span><span class="token punctuation">]</span> args<span class="token punctuation">)</span> <span class="token punctuation">{</span>
        <span class="token function">sort</span><span class="token punctuation">(</span>arr<span class="token punctuation">)</span><span class="token punctuation">;</span>
    <span class="token punctuation">}</span>

    <span class="token keyword">public</span> <span class="token keyword">static</span> <span class="token keyword">void</span> <span class="token function">sort</span><span class="token punctuation">(</span><span class="token keyword">int</span><span class="token punctuation">[</span><span class="token punctuation">]</span> arr<span class="token punctuation">)</span> <span class="token punctuation">{</span>
        <span class="token keyword">int</span> insertVal <span class="token operator">=</span> <span class="token number">0</span><span class="token punctuation">;</span>
        <span class="token keyword">int</span> insertIndex <span class="token operator">=</span> <span class="token number">0</span><span class="token punctuation">;</span>
        <span class="token keyword">for</span> <span class="token punctuation">(</span><span class="token keyword">int</span> i <span class="token operator">=</span> <span class="token number">1</span><span class="token punctuation">;</span> i <span class="token operator">&lt;</span> arr<span class="token punctuation">.</span>length<span class="token punctuation">;</span> i<span class="token operator">++</span><span class="token punctuation">)</span> <span class="token punctuation">{</span>
            insertVal <span class="token operator">=</span> arr<span class="token punctuation">[</span>i<span class="token punctuation">]</span><span class="token punctuation">;</span>
            insertIndex <span class="token operator">=</span> i<span class="token operator">-</span><span class="token number">1</span><span class="token punctuation">;</span>
            <span class="token keyword">while</span> <span class="token punctuation">(</span>insertIndex<span class="token operator">>=</span><span class="token number">0</span><span class="token operator">&amp;&amp;</span>insertVal<span class="token operator">&lt;</span>arr<span class="token punctuation">[</span>insertIndex<span class="token punctuation">]</span><span class="token punctuation">)</span><span class="token punctuation">{</span>
                arr<span class="token punctuation">[</span>insertIndex<span class="token operator">+</span><span class="token number">1</span><span class="token punctuation">]</span> <span class="token operator">=</span> arr<span class="token punctuation">[</span>insertIndex<span class="token punctuation">]</span><span class="token punctuation">;</span>
                insertIndex<span class="token operator">--</span><span class="token punctuation">;</span>
            <span class="token punctuation">}</span>
            <span class="token keyword">if</span> <span class="token punctuation">(</span>insertIndex<span class="token operator">!=</span>i<span class="token operator">-</span><span class="token number">1</span><span class="token punctuation">)</span><span class="token punctuation">{</span>
                arr<span class="token punctuation">[</span>insertIndex<span class="token operator">+</span><span class="token number">1</span><span class="token punctuation">]</span> <span class="token operator">=</span> insertVal<span class="token punctuation">;</span>
            <span class="token punctuation">}</span>
        <span class="token punctuation">}</span>
        <span class="token keyword">for</span> <span class="token punctuation">(</span><span class="token keyword">int</span> i <span class="token operator">=</span> <span class="token number">0</span><span class="token punctuation">;</span> i <span class="token operator">&lt;</span>arr<span class="token punctuation">.</span>length <span class="token punctuation">;</span> i<span class="token operator">++</span><span class="token punctuation">)</span> <span class="token punctuation">{</span>
            System<span class="token punctuation">.</span>out<span class="token punctuation">.</span><span class="token function">println</span><span class="token punctuation">(</span>arr<span class="token punctuation">[</span>i<span class="token punctuation">]</span><span class="token punctuation">)</span><span class="token punctuation">;</span>
        <span class="token punctuation">}</span>
    <span class="token punctuation">}</span>
<span class="token punctuation">}</span>
</code></pre>
<h3 id="4、希尔排序（缩小增量排序）"><a href="#4、希尔排序（缩小增量排序）" class="headerlink" title="4、希尔排序（缩小增量排序）"></a>4、希尔排序（缩小增量排序）</h3><h4 id="4-1、简单插入排序存在的问题"><a href="#4-1、简单插入排序存在的问题" class="headerlink" title="4.1、简单插入排序存在的问题"></a>4.1、简单插入排序存在的问题</h4><p>我们看简单的插入排序可能存在的问题.<br>数组 arr = {2,3,4,5,6,1} 这时需要插入的数 1(最小), 这样的过程是：</p>
<p>{2,3,4,5,6,6}<br>{2,3,4,5,5,6}<br>{2,3,4,4,5,6}<br>{2,3,3,4,5,6}<br>{2,2,3,4,5,6}<br>{1,2,3,4,5,6}</p>
<p><strong>结论：</strong>当需要插入的数是较小的数时，后移的次数明显增多，对效率有影响</p>
<h4 id="4-2、希尔排序法的基本思想"><a href="#4-2、希尔排序法的基本思想" class="headerlink" title="4.2、希尔排序法的基本思想"></a>4.2、希尔排序法的基本思想</h4><p>希尔排序是把记录按下标的一定增量分组，对每组使用直接插入排序算法排序；随着增量逐渐减少，每组包含的关键词越来越多，当增量减至 1 时，整个文件恰被分成一组，算法便终止</p>
<h4 id="4-3、希尔排序法的示意图"><a href="#4-3、希尔排序法的示意图" class="headerlink" title="4.3、希尔排序法的示意图"></a>4.3、希尔排序法的示意图</h4><p><img src="https://notes-hub.oss-cn-beijing.aliyuncs.com/typora_images/image-20220904225233559.png" alt="image-20220904225233559"></p>
<p><img src="https://notes-hub.oss-cn-beijing.aliyuncs.com/typora_images/image-20220904225247869.png" alt="image-20220904225247869"></p>
<h4 id="4-4、希尔排序法应用实例"><a href="#4-4、希尔排序法应用实例" class="headerlink" title="4.4、希尔排序法应用实例"></a>4.4、希尔排序法应用实例</h4><blockquote>
<p>有一群小牛, 考试成绩分别是 {8,9,1,7,2,3,5,4,6,0} 请从小到大排序</p>
</blockquote>
<pre class=" language-java"><code class="language-java"><span class="token comment" spellcheck="true">/**
 * 有一群小牛, 考试成绩分别是 {8,9,1,7,2,3,5,4,6,0} 请从小到大排序
 */</span>
<span class="token keyword">public</span> <span class="token keyword">class</span> <span class="token class-name">ShellSorting</span> <span class="token punctuation">{</span>
    <span class="token keyword">static</span> <span class="token keyword">int</span><span class="token punctuation">[</span><span class="token punctuation">]</span> arr <span class="token operator">=</span> <span class="token punctuation">{</span><span class="token number">8</span><span class="token punctuation">,</span><span class="token number">9</span><span class="token punctuation">,</span><span class="token number">1</span><span class="token punctuation">,</span><span class="token number">7</span><span class="token punctuation">,</span><span class="token number">2</span><span class="token punctuation">,</span><span class="token number">3</span><span class="token punctuation">,</span><span class="token number">5</span><span class="token punctuation">,</span><span class="token number">4</span><span class="token punctuation">,</span><span class="token number">6</span><span class="token punctuation">,</span><span class="token number">0</span><span class="token punctuation">}</span><span class="token punctuation">;</span>

    <span class="token keyword">public</span> <span class="token keyword">static</span> <span class="token keyword">void</span> <span class="token function">main</span><span class="token punctuation">(</span>String<span class="token punctuation">[</span><span class="token punctuation">]</span> args<span class="token punctuation">)</span> <span class="token punctuation">{</span>
        <span class="token comment" spellcheck="true">//交换法</span>
        <span class="token function">sort</span><span class="token punctuation">(</span>arr<span class="token punctuation">)</span><span class="token punctuation">;</span>
        System<span class="token punctuation">.</span>out<span class="token punctuation">.</span><span class="token function">println</span><span class="token punctuation">(</span><span class="token punctuation">)</span><span class="token punctuation">;</span>
        <span class="token comment" spellcheck="true">//移位法</span>
        <span class="token function">sort2</span><span class="token punctuation">(</span>arr<span class="token punctuation">)</span><span class="token punctuation">;</span>
    <span class="token punctuation">}</span>
    <span class="token comment" spellcheck="true">/**
     * 交换法
     */</span>
    <span class="token keyword">public</span> <span class="token keyword">static</span> <span class="token keyword">void</span> <span class="token function">sort</span><span class="token punctuation">(</span><span class="token keyword">int</span><span class="token punctuation">[</span><span class="token punctuation">]</span> arr<span class="token punctuation">)</span><span class="token punctuation">{</span>
        <span class="token keyword">int</span> temp <span class="token operator">=</span> <span class="token number">0</span><span class="token punctuation">;</span>
        <span class="token keyword">for</span> <span class="token punctuation">(</span><span class="token keyword">int</span> grap <span class="token operator">=</span> arr<span class="token punctuation">.</span>length<span class="token operator">/</span><span class="token number">2</span><span class="token punctuation">;</span> grap<span class="token operator">></span><span class="token number">0</span><span class="token punctuation">;</span>grap <span class="token operator">/=</span> <span class="token number">2</span><span class="token punctuation">)</span> <span class="token punctuation">{</span>
            <span class="token keyword">for</span> <span class="token punctuation">(</span><span class="token keyword">int</span> i <span class="token operator">=</span> grap<span class="token punctuation">;</span> i <span class="token operator">&lt;</span> arr<span class="token punctuation">.</span>length<span class="token punctuation">;</span> i<span class="token operator">++</span><span class="token punctuation">)</span> <span class="token punctuation">{</span>
                <span class="token keyword">for</span> <span class="token punctuation">(</span><span class="token keyword">int</span> j <span class="token operator">=</span> i<span class="token operator">-</span>grap<span class="token punctuation">;</span> j <span class="token operator">>=</span><span class="token number">0</span> <span class="token punctuation">;</span> j<span class="token operator">-=</span>grap<span class="token punctuation">)</span> <span class="token punctuation">{</span>
                    <span class="token keyword">if</span> <span class="token punctuation">(</span>arr<span class="token punctuation">[</span>j<span class="token punctuation">]</span><span class="token operator">></span>arr<span class="token punctuation">[</span>j<span class="token operator">+</span>grap<span class="token punctuation">]</span><span class="token punctuation">)</span><span class="token punctuation">{</span>
                        temp <span class="token operator">=</span> arr<span class="token punctuation">[</span>j<span class="token punctuation">]</span><span class="token punctuation">;</span>
                        arr<span class="token punctuation">[</span>j<span class="token punctuation">]</span> <span class="token operator">=</span> arr<span class="token punctuation">[</span>j<span class="token operator">+</span>grap<span class="token punctuation">]</span><span class="token punctuation">;</span>
                        arr<span class="token punctuation">[</span>j<span class="token operator">+</span>grap<span class="token punctuation">]</span> <span class="token operator">=</span> temp<span class="token punctuation">;</span>
                    <span class="token punctuation">}</span>
                <span class="token punctuation">}</span>
            <span class="token punctuation">}</span>
        <span class="token punctuation">}</span>
        <span class="token keyword">for</span> <span class="token punctuation">(</span><span class="token keyword">int</span> i <span class="token operator">=</span> <span class="token number">0</span><span class="token punctuation">;</span> i <span class="token operator">&lt;</span> arr<span class="token punctuation">.</span>length<span class="token punctuation">;</span> i<span class="token operator">++</span><span class="token punctuation">)</span> <span class="token punctuation">{</span>
            System<span class="token punctuation">.</span>out<span class="token punctuation">.</span><span class="token function">print</span><span class="token punctuation">(</span>arr<span class="token punctuation">[</span>i<span class="token punctuation">]</span><span class="token operator">+</span><span class="token string">"-"</span><span class="token punctuation">)</span><span class="token punctuation">;</span>
        <span class="token punctuation">}</span>
    <span class="token punctuation">}</span>

    <span class="token comment" spellcheck="true">/**
     * 移位法
     * @param arr
     */</span>
    <span class="token keyword">public</span> <span class="token keyword">static</span> <span class="token keyword">void</span> <span class="token function">sort2</span><span class="token punctuation">(</span><span class="token keyword">int</span><span class="token punctuation">[</span><span class="token punctuation">]</span> arr<span class="token punctuation">)</span><span class="token punctuation">{</span>
        <span class="token keyword">for</span> <span class="token punctuation">(</span><span class="token keyword">int</span> grap <span class="token operator">=</span> arr<span class="token punctuation">.</span>length<span class="token operator">/</span><span class="token number">2</span><span class="token punctuation">;</span>grap <span class="token operator">></span> <span class="token number">0</span> <span class="token punctuation">;</span> grap<span class="token operator">/=</span><span class="token number">2</span><span class="token punctuation">)</span> <span class="token punctuation">{</span>
            <span class="token keyword">for</span> <span class="token punctuation">(</span><span class="token keyword">int</span> i <span class="token operator">=</span> grap<span class="token punctuation">;</span> i <span class="token operator">&lt;</span>arr<span class="token punctuation">.</span>length <span class="token punctuation">;</span> i<span class="token operator">++</span><span class="token punctuation">)</span> <span class="token punctuation">{</span>
                <span class="token keyword">int</span> j <span class="token operator">=</span> i<span class="token punctuation">;</span>
                <span class="token keyword">int</span> temp <span class="token operator">=</span> arr<span class="token punctuation">[</span>j<span class="token punctuation">]</span><span class="token punctuation">;</span>
                <span class="token keyword">if</span> <span class="token punctuation">(</span>arr<span class="token punctuation">[</span>j<span class="token punctuation">]</span><span class="token operator">&lt;</span>arr<span class="token punctuation">[</span>j<span class="token operator">-</span>grap<span class="token punctuation">]</span><span class="token punctuation">)</span><span class="token punctuation">{</span>
                    <span class="token keyword">while</span> <span class="token punctuation">(</span>j<span class="token operator">-</span>grap<span class="token operator">>=</span><span class="token number">0</span><span class="token operator">&amp;&amp;</span>temp<span class="token operator">&lt;</span>arr<span class="token punctuation">[</span>j<span class="token operator">-</span>grap<span class="token punctuation">]</span><span class="token punctuation">)</span><span class="token punctuation">{</span>
                        arr<span class="token punctuation">[</span>j<span class="token punctuation">]</span> <span class="token operator">=</span> arr<span class="token punctuation">[</span>j<span class="token operator">-</span>grap<span class="token punctuation">]</span><span class="token punctuation">;</span>
                        j <span class="token operator">-=</span> grap<span class="token punctuation">;</span>
                    <span class="token punctuation">}</span>
                    arr<span class="token punctuation">[</span>j<span class="token punctuation">]</span> <span class="token operator">=</span> temp<span class="token punctuation">;</span>
                <span class="token punctuation">}</span>
            <span class="token punctuation">}</span>
        <span class="token punctuation">}</span>

        <span class="token keyword">for</span> <span class="token punctuation">(</span><span class="token keyword">int</span> i <span class="token operator">=</span> <span class="token number">0</span><span class="token punctuation">;</span> i <span class="token operator">&lt;</span> arr<span class="token punctuation">.</span>length<span class="token punctuation">;</span> i<span class="token operator">++</span><span class="token punctuation">)</span> <span class="token punctuation">{</span>
            System<span class="token punctuation">.</span>out<span class="token punctuation">.</span><span class="token function">print</span><span class="token punctuation">(</span>arr<span class="token punctuation">[</span>i<span class="token punctuation">]</span><span class="token operator">+</span><span class="token string">"-"</span><span class="token punctuation">)</span><span class="token punctuation">;</span>
        <span class="token punctuation">}</span>
    <span class="token punctuation">}</span>

<span class="token punctuation">}</span>
</code></pre>
<h3 id="5、快速排序"><a href="#5、快速排序" class="headerlink" title="5、快速排序"></a>5、快速排序</h3><h4 id="5-1、快速排序法介绍"><a href="#5-1、快速排序法介绍" class="headerlink" title="5.1、快速排序法介绍"></a>5.1、快速排序法介绍</h4><p>快速排序（Quicksort）是对冒泡排序的一种改进。基本思想是：通过一趟排序将要排序的数据分割成独立的两部分，其中一部分的所有数据都比另外一部分的所有数据都要小，然后再按此方法对这两部分数据分别进行快速排<br>序，整个排序过程可以递归进行，以此达到整个数据变成有序序列</p>
<h4 id="5-2、快速排序法示意图"><a href="#5-2、快速排序法示意图" class="headerlink" title="5.2、快速排序法示意图"></a>5.2、快速排序法示意图</h4><p><img src="https://notes-hub.oss-cn-beijing.aliyuncs.com/typora_images/image-20220905160314804.png" alt="image-20220905160314804"></p>
<h4 id="5-3、快速排序法应用实例"><a href="#5-3、快速排序法应用实例" class="headerlink" title="5.3、快速排序法应用实例"></a>5.3、快速排序法应用实例</h4><p>要求: 对 [-9,78,0,23,-567,70] 进行从小到大的排序，要求使用快速排序法。</p>
<ol>
<li><p>如果取消左右递归，结果是 -9 -567 0 23 78 70</p>
</li>
<li><p>如果取消右递归,结果是 -567 -9 0 23 78 70</p>
</li>
<li><p>如果取消左递归,结果是 -9 -567 0 23 70 78</p>
</li>
</ol>
<p><strong>coding：</strong></p>
<pre class=" language-java"><code class="language-java"><span class="token comment" spellcheck="true">/**
 * 对 [-9,78,0,23,-567,70] 进行从小到大的排序，要求使用快速排序法。
 */</span>
<span class="token keyword">public</span> <span class="token keyword">class</span> <span class="token class-name">QuickSorting</span> <span class="token punctuation">{</span>
    <span class="token keyword">static</span> <span class="token keyword">int</span><span class="token punctuation">[</span><span class="token punctuation">]</span> arr <span class="token operator">=</span> <span class="token punctuation">{</span><span class="token operator">-</span><span class="token number">9</span><span class="token punctuation">,</span><span class="token number">78</span><span class="token punctuation">,</span><span class="token number">0</span><span class="token punctuation">,</span><span class="token number">23</span><span class="token punctuation">,</span><span class="token operator">-</span><span class="token number">567</span><span class="token punctuation">,</span><span class="token number">70</span><span class="token punctuation">}</span><span class="token punctuation">;</span>

    <span class="token keyword">public</span> <span class="token keyword">static</span> <span class="token keyword">void</span> <span class="token function">sort</span><span class="token punctuation">(</span><span class="token keyword">int</span><span class="token punctuation">[</span><span class="token punctuation">]</span> arr<span class="token punctuation">,</span><span class="token keyword">int</span> left<span class="token punctuation">,</span> <span class="token keyword">int</span> right<span class="token punctuation">)</span><span class="token punctuation">{</span>
        <span class="token keyword">int</span> l <span class="token operator">=</span> left<span class="token punctuation">;</span>
        <span class="token keyword">int</span> r <span class="token operator">=</span> right<span class="token punctuation">;</span>
        <span class="token keyword">int</span> pivot <span class="token operator">=</span> arr<span class="token punctuation">[</span><span class="token punctuation">(</span>left<span class="token operator">+</span>right<span class="token punctuation">)</span><span class="token operator">/</span><span class="token number">2</span><span class="token punctuation">]</span><span class="token punctuation">;</span>
        <span class="token keyword">int</span> temp <span class="token operator">=</span> <span class="token number">0</span><span class="token punctuation">;</span>
        <span class="token keyword">while</span> <span class="token punctuation">(</span>l<span class="token operator">&lt;</span>r<span class="token punctuation">)</span><span class="token punctuation">{</span>
            <span class="token keyword">while</span> <span class="token punctuation">(</span>arr<span class="token punctuation">[</span>l<span class="token punctuation">]</span><span class="token operator">&lt;</span>pivot<span class="token punctuation">)</span><span class="token punctuation">{</span>
                l<span class="token operator">+=</span><span class="token number">1</span><span class="token punctuation">;</span>
            <span class="token punctuation">}</span>
            <span class="token keyword">while</span> <span class="token punctuation">(</span>arr<span class="token punctuation">[</span>r<span class="token punctuation">]</span><span class="token operator">></span>pivot<span class="token punctuation">)</span><span class="token punctuation">{</span>
                r<span class="token operator">+=</span><span class="token number">1</span><span class="token punctuation">;</span>
            <span class="token punctuation">}</span>

            <span class="token keyword">if</span> <span class="token punctuation">(</span>l<span class="token operator">>=</span>r<span class="token punctuation">)</span><span class="token punctuation">{</span>
                <span class="token comment" spellcheck="true">//说明pivot左边全是比pivot小的，右边全是比pivot大的</span>
                <span class="token keyword">break</span><span class="token punctuation">;</span>
            <span class="token punctuation">}</span>
            temp <span class="token operator">=</span> arr<span class="token punctuation">[</span>l<span class="token punctuation">]</span><span class="token punctuation">;</span>
            arr<span class="token punctuation">[</span>l<span class="token punctuation">]</span> <span class="token operator">=</span> arr<span class="token punctuation">[</span>r<span class="token punctuation">]</span><span class="token punctuation">;</span>
            arr<span class="token punctuation">[</span>r<span class="token punctuation">]</span> <span class="token operator">=</span> temp<span class="token punctuation">;</span>
        <span class="token punctuation">}</span>
        <span class="token keyword">if</span> <span class="token punctuation">(</span>arr<span class="token punctuation">[</span>l<span class="token punctuation">]</span> <span class="token operator">==</span> pivot<span class="token punctuation">)</span><span class="token punctuation">{</span>
            l<span class="token operator">-=</span><span class="token number">1</span><span class="token punctuation">;</span>
        <span class="token punctuation">}</span>
        <span class="token keyword">if</span> <span class="token punctuation">(</span>arr<span class="token punctuation">[</span>r<span class="token punctuation">]</span> <span class="token operator">==</span> pivot<span class="token punctuation">)</span><span class="token punctuation">{</span>
            r<span class="token operator">+=</span><span class="token number">1</span><span class="token punctuation">;</span>
        <span class="token punctuation">}</span>
        <span class="token keyword">if</span> <span class="token punctuation">(</span>left<span class="token operator">&lt;</span>l<span class="token punctuation">)</span><span class="token punctuation">{</span>
            <span class="token function">sort</span><span class="token punctuation">(</span>arr<span class="token punctuation">,</span>left<span class="token punctuation">,</span>l<span class="token punctuation">)</span><span class="token punctuation">;</span>
        <span class="token punctuation">}</span>
        <span class="token keyword">if</span> <span class="token punctuation">(</span>right<span class="token operator">></span>r<span class="token punctuation">)</span><span class="token punctuation">{</span>
            <span class="token function">sort</span><span class="token punctuation">(</span>arr<span class="token punctuation">,</span>r<span class="token punctuation">,</span>right<span class="token punctuation">)</span><span class="token punctuation">;</span>
        <span class="token punctuation">}</span>
    <span class="token punctuation">}</span>
<span class="token punctuation">}</span>
</code></pre>
<h3 id="6、归并排序"><a href="#6、归并排序" class="headerlink" title="6、归并排序"></a>6、归并排序</h3><h4 id="6-1、归并排序介绍"><a href="#6-1、归并排序介绍" class="headerlink" title="6.1、归并排序介绍"></a>6.1、归并排序介绍</h4><p>归并排序（MERGE-SORT）是利用归并的思想实现的排序方法，该算法采用经典的分治（divide-and-conquer）策略（分治法将问题分(divide)成一些小的问题然后递归求解，而治(conquer)的阶段则将分的阶段得到的各答案”修<br>补”在一起，即分而治之)</p>
<h4 id="6-2、归并排序思想示意图1-基本思想"><a href="#6-2、归并排序思想示意图1-基本思想" class="headerlink" title="6.2、归并排序思想示意图1-基本思想"></a>6.2、归并排序思想示意图1-基本思想</h4><p><img src="https://notes-hub.oss-cn-beijing.aliyuncs.com/typora_images/image-20220905170453288.png" alt="image-20220905170453288"></p>
<h4 id="6-3、归并排序思想示意图-2-合并相邻有序子序列"><a href="#6-3、归并排序思想示意图-2-合并相邻有序子序列" class="headerlink" title="6.3、归并排序思想示意图 2-合并相邻有序子序列"></a>6.3、归并排序思想示意图 2-合并相邻有序子序列</h4><p>再来看看治阶段，我们需要将两个已经有序的子序列合并成一个有序序列，比如上图中的最后一次合并，要将[4,5,7,8]和[1,2,3,6]两个已经有序的子序列，合并为最终序列[1,2,3,4,5,6,7,8]，来看下实现步骤</p>
<p><img src="https://notes-hub.oss-cn-beijing.aliyuncs.com/typora_images/image-20220905170618695.png" alt="image-20220905170618695"></p>
<h4 id="6-4、归并排序的应用实例"><a href="#6-4、归并排序的应用实例" class="headerlink" title="6.4、归并排序的应用实例"></a>6.4、归并排序的应用实例</h4><p>给你一个数组, val arr = Array(8, 4, 5, 7, 1, 3, 6, 2 ), 请使用归并排序完成排序</p>

                
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